Let abc be a three-digit number. If all six substrings of abc are prime, then we must have:

a, b, c are prime

ab, bc are prime

abc is prime

It follows that a, b and c must each be one of the values 2, 3, 5, or 7. There are thus 16 potential values for each of ab and bc. However, only four of them are prime: 23, 37, 53, or 73. From these four values we have four potential values for abc: 237, 373, 537 and 737. Only of them is prime: 373.